27371 is a member of OEIS A001607: a(n) = -a(n-1) - 2*a(n-2). The sequence can be generated as follows:

INPUT

a,b=1,1

L=[]

for n in [1..31]:

  c=-b-2*a

  L.append(c)

  a,b=b,c

print(L)

list_plot(L,plotjoined=True)

OUTPUT

[-3, 1, 5, -7, -3, 17, -11, -23, 45, 1, -91, 89, 93, -271, 85, 457, -627, -287, 1541, -967, -2115, 4049, 181, -8279, 7917, 8641, -24475, 7193, 41757, -56143, -27371]

Here are some relevant OEIS comments:

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

Apart from the sign, this is an example of a sequence of Lehmer numbers. In this case, the two parameters, alpha and beta, are (1 +- i*sqrt(7))/2. Bilu, Hanrot, Voutier and Mignotte show that all terms of a Lehmer sequence a(n) have a primitive factor for n > 30.

Note that for this sequence, a(30) = 24475 = 5 * 5 * 11 * 89 has no primitive factors. 

27371 is a cyclic and Duffinian number.

27371 stabilises after over 880 generations into a complex mix of still lifes, blinkers and traffic lights.